A top of this kind is known from practice and can be designed as a folding top, which has a foldable top cover, or as a retractable hard top, which has rigid top shells. The top comprises a top linkage, by means of which, depending on the design, the top cover can be stretched or the rigid top shells can be adjusted. The top linkage is adjustable between a closed position, in which a vehicle interior of the vehicle in question is covered, and a closed position, in which the vehicle interior is open to the top and the top linkage is stored in a rear-side top storage space of the vehicle in question. The top storage space can typically be closed by means of an adjustable storage space lid, which can form a hat shelf arranged in the vehicle interior when the top is in the closed position and which is pivotably mounted on a vehicle-fixed main bearing via a link mechanism on each of its two sides with respect to a vertical longitudinal center plane of the top. The top linkage has a linkage mechanism on each of its two sides with respect to the vertical longitudinal center plane of the top, each linkage mechanism being pivotably mounted on the respective vehicle-fixed main bearing, as well, and being drivable by means of a main drive arranged on the main bearing. For adjusting the storage space lid, a separate drive motor has been provided so far, which causes costs, on the one hand, and requires a large amount of installation space, on the other hand.